Analytic Connectivity in General Hypergraphs

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by  Ashwin Guha

ANALYTIC CONNECTIVITY IN GENERAL
HYPERGRAPHS
ASHWIN GUHA, MUNI SREENIVAS PYDI, BISWAJIT PARIA,
AND AMBEDKAR DUKKIPATI
Abstract. Inthis paper we extendthe knownresultsof analytic
7
1 connectivity to non-uniform hypergraphs. We prove a modified
0 Cheeger’sinequalityandalsogiveaboundonanalyticconnectivity
2 withrespecttothedegreesequenceanddiameterofahypergraph.

n
a
J

7
1
1. Introduction
]
M
Combinatorial graph theory is a vast active area of research with a
D
wide variety of applications. It is well-known that many interesting
.
s problems have no polynomial-time algorithms. In an attempt to find
c
approximate solutions to these problems, matrices pertaining to the
[

graph such as adjacency matrix or Laplacian matrix were studied, and
1
v the eigenvalues of these matrices were used to bound various graph
8 parameters. This gave rise to spectral graph theory, which has since
4
become a separate area of research in its own right.
5
4 The idea of graphs has been generalized to hypergraphs where an
0 edge may span more than two vertices. Hypergraphs have also been
.
1 explored in detail [2]. However, spectral methods for hypergraphs have
0
received less attention. Recently there has been a renewed interest in
7
1 spectral hypergraph theory.
:
v Thetraditional approachtowardsdealing with hypergraphs hasbeen
i to convert it into a graph and apply the known graph results. Another
X
approach is to define a tensor for a hypergraph that is an analogue of
r
a
the corresponding matrix for a graph. A novel definition of eigenvalue
for a tensor given by Qi [8] and independently by Lim [7] has led to
surge of activity in this area. Subsequently various results on spectra
of hypergraphs have been provided in [9, 4] . The results in these works
are applicable to uniform hypergraphs where the size of every edge is
fixed. There have been recent attempts to extend these results for
non-uniform hypergraphs [1, 3].
The concept of analytic connectivity was introduced in [9], which
is the analogue of algebraic connectivity introduced by Fiedler [5] for
graphs. In [6], the relations between analytic connectivity and other
graph parameters are explored. In particular, Cheeger’s inequality for
uniform hypergraphs is proved.
1
2 ANALYTIC CONNECTIVITY IN GENERAL HYPERGRAPHS
In this paper we use the terminology given by Banerjee et al. [1]
and extend the results by Li et al. [6] to non-uniform hypergraphs.
We prove a modified Cheeger’s inequality and also give the relation
between analytic connectivity and diameter.
The paper is organized as follows. In Section 2 we introduce our
notation and state the preliminaries. We state the equivalent theorems
for graphs and uniform hypergraphs. In Section 3, we provide the main
results with proof. Section 4 contains concluding remarks.
2. Preliminaries
Let H(V,E) be a general hypergraph on vertex set V and edge set E
which is a set of non-empty subsets of V. Let n be number of vertices.
We denote by m and s the size of the largest edge and the smallest
min
edge respectively, i.e., m = max e : e E and s = min e : e
min
{| | ∈ } {| | ∈
E . Let ∆ be the largest degree among vertices of H.
}
For a hypergraph H we adopt the definition given in [1] and define
the adjacency tensor to be an n-dimensional hypermatrix 1 of order
A
msuchthatife = v ,v ,...,v E isanedgeofcardinalitys m,
{ i1 i2 is} ∈ ≤
then
s m!
a = , where Ω =
p1p2...pm Ω k !k !...k !
1 2 s
kXj≥1
with k = m and p ,p ,...,p are chosen in all possible ways from
j 1 2 m
i ,i ,...,i such that each element of the set appears at least once.
1 2 s
{ P }
The other entries of the hypermatrix are zero. Note that this definition
agrees with the definition of adjacency tensor for uniform hypergraphs
given in [4].
For an edge e = v ,v ,...,v E we define
{ i1 i2 is} ∈
xe = x x ...x ,
m r1 r2 rm
X
where the sum is over r ,r ,...,r chosen in all possible ways from
1 2 m
i ,i ,...,i with each element of the set appearing at least once in
1 2 s
{ }
the index .
Let be the degree tensor which is a diagonal tensor of order m
D
and dimension n such that d = d(v ) and zero elsewhere. We define
ii...i i
Laplacian tensor (H) to be .
For a tensor Lof order mDan−dAdimension n and x Rn we define
T ∈
xm as
T
n
( xm) = t x x ...x .
T j ji2i3...im i2 i3 im
i2,.X..,im=1
1Inthis paperweuse the termshypermatrixandtensorinterchangeablyforeaseof
understanding.
ANALYTIC CONNECTIVITY IN GENERAL HYPERGRAPHS 3
In particular we have
n n
xm = ( )xm = d(v )xm a x ...x
L D −A i i − ii2...im i2 im
Xi=1 i2,.X..,im=1
s
s
= xm xe = (e)xm
ij − Ω m L
!
e={vi1X,...,vis}∈E Xj=1 Xe∈E
s
s
where (e)xm = xm xe .
L ij − Ω m
j=1
It can be shownXthat for x Rn, (e)xm 0 [1]. We recall the
∈ + L ≥
definitions of a few hypergraph parameters which are intuitive gener-
alizations of those for a 2-graph.
Definition 2.1. For a hypergraph H, the isoperimetric number i(H)
is defined as
∂S V
i(H) = min | | : S V,0 S | | ,
S ⊂ ≤ | | ≤ 2
(cid:26) | | (cid:27)
where ∂S is the boundary of S which consists of the edges in H with
vertices in both S and S = V S. When H is a 2-graph, the edges in
\
∂S have exactly one vertex in S and one vertex in S.
Definition 2.2. For a hypergraph H, the diameter diam(H) is defined
as the maximum distance between any pair of vertices.
diam(H) = max l(u,v) : u,v V ,
{ ∈ }
where l(u,v) is the length of the shortest path connecting u and v.
The concept of algebraic connectivity, defined as the second smallest
eigenvalue of the Laplacian (denoted by λ ) was introduced by Fiedler
2
[5]. The algebraic connectivity has proved to be a reliable measure to
understand the structure of a 2-graph. Some of the results which give
bounds on λ with respect to various graph constants are given below,
2
including a version of the famous Cheeger inequality.
Lemma 2.3. For a graph G with n vertices,
(x x )2
λ (G) = 2nmin (vi,vj)∈E i − j : x = c.1 , for all c R .
2 n n (x x )2 6 n ∈
( Pi=1 j=1 i − j )
Theorem 2.4. ForPa graPph G with n vertices,
4
λ (G) .
2 ≥ n diam(G)
·
Theorem 2.5. LetG be a 2-graphwith more thanone edgeand(d(v ),...,d(v ))
1 n
be the degree sequence of G. Then
d(v )+d(v ) 2
i j
λ (G) min − .
2
≤ {vi,vj}∈E(cid:26) 2 (cid:27)
4 ANALYTIC CONNECTIVITY IN GENERAL HYPERGRAPHS
Theorem 2.6. For a graph G,
2i(G) λ (G) ∆(G) ∆(G)2 i(G)2.
2
≥ ≥ − −
For k-graphs the concept of analytic cponnectivity was introduced by
Qi [9] as the equivalent of λ . The definition is also valid for non-
2
uniform hypergraphs as mentioned in [1].
Definition 2.7. The analytic connectivity of a k-uniform hypergraph
H is defined as
n
α(H) = min min xm : x Rn, xm = 1,x = 0 .
j=1,...,n {L ∈ + i j }
i=1
X
The above results for 2-graphs have been extended to k-graphs in
[6] where the inequalities are presented with respect to α(H). The
theorems are given below.
Theorem 2.8. Let H be a k-graph. Then
4
α(H) .
≥ n2(k 1)diam(H)
−
Theorem 2.9. Let H be k-graph with more than one edge. Then
d(v )+d(v )+...+d(v ) k
α(H) min i1 i2 ik − : v ,v ,...,v E .
≤ k { i1 i2 ik} ∈
(cid:26) (cid:27)
Theorem 2.10. For a k-graph H with k 3,
≥
k
i α(H) ∆ √∆2 i2.
2 ≥ ≥ − −
We also mention a lemma from [6] which will be used to prove the
main results in the next section.
Lemma 2.11. Let a = (a ,...,a ) Rn. Let A = (a +...+ a )/n
1 n ∈ + 1 n
and G = (a ...a )1/n. Then
1 n
1
A G (√a √a )2. (1)
i j
− ≥ n(n 1) −
− 1≤i<j≤n
X
⌊n/2⌋
1
A G ( b b )2. (2)
j n+1−j
− ≥ n −
j=1
X p p
where b = a , for j = 1,...,n and σ is a permutation of the set
j σ(j)
1,...,n .
{ }
3. Results for general hypergraphs
Inthissectionweprovethecorrespondingresultsfornon-uniformhy-
pergraphs. The inequalities are similar to that of uniform hypergraphs,
except for an additional factor of smin. The proofs can be obtained by
m
modifying those in [6]. We have given the detailed proofs here for the
sake of clarity and completeness.
ANALYTIC CONNECTIVITY IN GENERAL HYPERGRAPHS 5
Theorem 3.1. Let H be a general hypergraph. Then
4smin
α(H) .
≥ n2m(m 1)diam(H)
−
Proof. Let x = (x ,x ,...,x ) be the vector achieving α(H). Assume
1 2 n
x = 0 and let y = x(m/2). Define a 2-graph H∗ with vertex set V(H)
n
and u v in H∗ if and only if u,v e E. In other words H∗ is
∼ { } ⊂ ∈
the clique expansion of H. We know that diam(H) = diam(H∗). For
edge e = v ,...,v E , consider k copies of xm, k copies of xm
{ i1 is} ∈ 1 1 2 2
and k copies of xm, where k 1 and k = m. From Lemma 2.11
s s j ≥ j
we have,
P
k xm +k xm +...k xm
1 1 2 2 s s xk1xk2 ...xks
m − 1 2 s
1
[(xm/2 xm/2)2 +(xm/2 xm/2)2 +...+(xm/2 xm/2)2]
≥ m(m 1) 1 − 2 1 − 3 s−1 − s
−
1
= (xm/2 xm/2)2.
m(m 1) i − j
− 1≤i<j≤s
X
Summing over different values of k ,k ,...,k we get
1 2 s
s
Ω Ω
xm xe (xm/2 xm/2)2
s i − m ≥ m(m 1) i − j
!
i=1 − 1≤i<j≤s
X X
s
1 s 1
xm xe (xm/2 xm/2)2
s i − Ω m ≥ m(m 1) i − j
!
i=1 − 1≤i<j≤s
X X
s 1
(e)xm (xm/2 xm/2)2
L ≥ m(m 1) i − j
− 1≤i<j≤s
X
s 1
min (xm/2 xm/2)2.
≥ m (m 1) i − j
− 1≤i<j≤s
X
α = (e)xm
L
e∈E(H)
X
s 1
min (xm/2 xm/2)2
≥ m (m 1) i − j
− (vivjX)∈E(H∗)
s 1
= min (y y )2
i j
m (m 1) −
− (vivjX)∈E(H∗)
= smin 1 n n (y y )2 (vivj)∈E(H∗)(yi −yj)2
m (m 1) i − j n n (y y )2
− i=1 j=1 P i=1 j=1 i − j
XX
λ (H∗) s n n P P
2 min (y y )2 from Lemma 2.3.
i j
≥ 2n(m 1) m −
− i=1 j=1
XX
6 ANALYTIC CONNECTIVITY IN GENERAL HYPERGRAPHS
n n n n n n n n
(y y )2 = y2 + y2 2 y y
i − j i j − i j
i=1 j=1 i=1 j=1 i=1 j=1 i=1 j=1
XX XX XX XX
2
n n−1
= 2n y2 2 y (since y = 0)
i − i n
! !
i=1 i=1
X X
n
2n 2(n 1)( y2) (from Cauchy-Schwarz)
≥ − − i
i=1
X
= 2.
We have
λ (H∗) s
2 min
α 2.
≥ 2n(m 1) m ·
−
From Theorem 2.4, λ (H∗) 4 . Therefore,
2 ≥ diam(H∗)·n
s 4
min
α .
≥ m n2(m 1) diam(H∗)
− ·
(cid:3)
Theorem 3.2. Let H be a non-uniform hypergraph with more than
one edge. Then
d(v )+d(v )+...+d(v ) s
α(H) min i1 i2 is − : v ,v ,...,v E .
≤ s { i1 i2 is} ∈
(cid:26) (cid:27)
Proof. Let e = v ,v ,...,v E. Define a vector x Rn such
0 { i1 i2 is} ∈ ∈ +
that
s−m if v e
i 0
x = ∈
i
0 otherwise.
(
m m
1 s
Then xm = s( )m = 1. (e )xm = xm xe = 0.
i s1/m L 0 i − Ω m
i=1 i=1
X X
α(H) xm
≤ L
= (e)xm
L
e∈E
X
= (e)xm + (e )xm
0
 L L 
e∈EX\{e0}
= (d(v ) 1)(1/s)+(d(v ) 1)(1/s)+...+(d(v ) 1)(1/s)
i1 − i2 − is −
d(v )+d(v )+...+d(v ) s
= i1 i2 is − .
s
(cid:3)
Theorem 3.3. For a non-uniform hypergraph H,
m smin
i α(H) (∆ √∆2 i2).
2 ≥ ≥ m − −
ANALYTIC CONNECTIVITY IN GENERAL HYPERGRAPHS 7
Proof. Suppose S V gives the isoperimetric number i. Let y =
(y ,...,y ) Rn b⊂e the vector defined as follows.
1 n ∈ +
1 if v S
y = |S|1/m i ∈
i
(0 otherwise.
Let t (e) = v : v e S be the number of vertices of e in S and
S
|{ ∈ ∩ }|
t(S) = Pe∈∂StS(e).
|∂S|
t (e)+t (e)
t(S)+t(S) = e∈∂S S S m. (3)
∂S ≤
P | |
α ym = + + (e)ym. (4)
≤ L  L
Xe∈S Xe∈S eX∈∂S
 
Ife = v ,...,v S, (e)ym = ym sye = 0. Ife = v ,...,v
{ 1 s} ⊂ L i −Ω m { 1 s} ⊂
S,
P
m
s
(e)ym = ym ye
L i − Ω m
i=1
X
s s 1
= 1
S − Ω S ·
| | | |
= 0. X
Therefore only edges in the boundary contribute to the sum of (4).
α ym
≤ i
eX∈∂SviX∈e∩S
t (e) 1
S
= = t(S) ∂S
S S | |
e∈∂S | | | |
X
= t(S) i.
·
Similarly we can get α t(S) i. Adding them we get 2α (t(S) +
≤ · ≤
t(S))i. Combining with (3), we get α mi.
≤ 2
To prove the lower bound, suppose x = (x ,...,x ) achieves α.
1 n
For each edge e = v ,v ,...,v assume x x ... x by
{ i1 i2 is} i1 ≤ i2 ≤ ≤ is
rearranging the vertices. We define a 2-graph H whose vertex set is
same as H and edges are such that E(H) = v v : j =
∪e∈E(H){ ij is+1−j
1,..., s/2 . Then b
⌊ ⌋}
b
s
s
α = ( xm xk1 ...xks). (5)
ij − Ω i1 is
e={vi1X,vi2,...,vis} Xj=1 kXi≥1
Pki=m
8 ANALYTIC CONNECTIVITY IN GENERAL HYPERGRAPHS
Consider k copies of xm, k copies of xm and k copies of xm, where
1 1 2 2 s s
k 1 and k = m. Applying Lemma 2.11 we get
j j
≥
P
⌊m/2⌋
k xm +k xm +...k xm 1
1 1 2 2 s s xk1xk2 ...xks ( b b )2
m − 1 2 s ≥ m i − m+1−i
i=1
X p p
(6)
where b ,...,b is any permutation of the variables x . In particular
1 m i
considertheassignmentb = xm,b = xm,...,b = xm ,b =
1 i1 2 i2 ⌊s/2⌋ is/2 m+1−⌊s/2⌋
xm ,...,b = xm, and b ,...,b are assigned to any
is+1−⌊s/2⌋ m is ⌊s/2⌋+1 m−⌊s/2⌋
of the remaining m s variables. Then
−
⌊m/2⌋ ⌊s/2⌋
( b b )2 ( b b )2 ( since s m)
i m+1−i i m+1−i
− ≥ − ≤
i=1 i=1
X p p X p p
⌊s/2⌋
= ( xm xm )2
ij − is+1−j
j=1
X q q
Using the above inequality with (6), and summing over all possible
values of k ,...,k as in proof of Theorem 3.1 we get
1 s
⌊s/2⌋
s
α ( ( xm xm )2)
≥ m ij − is+1−j
e={vi1,..X.,vis}∈E(H) Xj=1 q q
⌊s/2⌋
s
min ( ( xm xm )2)
≥ m ij − is+1−j
e={vi1,..X.,vis}∈E(H) Xj=1 q q
s
= min ( xm xm)2).
m i − j
{vi,vXj}∈E(Hb) p q
Proceeding as in proof of k-graphs in [6], let M = (y y )2
{vi,vj}∈E′ i− j
where E′ = E(H) and y = √xm. From Cauchy-Schwarz inequality we
i i P
have
b
( y2 y2 )2
M {vi,vj}∈E′| i − j| . (7)
≥ (y +y )2
P{vi,vj}∈E′ i j
P
Let w (= 0) < w < ... < w be the distinct values of y , for
0 1 h i
i = 1,...,n. For j = 0,...,h, let V = v V : y w . For
j i i j
{ ∈ ≥ }
each edge e ∂(V ), let δ (e) = min V e , V e . Let δ(V ) =
j j j j j
∈ {| ∩ | | ∩ |}
min δ (e) : e ∂(V ) and δ(H) = min δ(V ).
j j j=0,...,h j
{ ∈ }
ANALYTIC CONNECTIVITY IN GENERAL HYPERGRAPHS 9
h h
y2 y2 = (y2 y2) = (w2 w2)
| i − j| i − j k − l
{vi,Xvj}∈E′ Xk=1{vi,Xvj}∈E′ Xk=1{vi,Xvj}∈E′
vvji∈∈/VVkk yyijl=<=wkwkl
h
= (w2 w2 )+(w2 w2 )+...+(w2 w2)
k − k−1 k−1− k−2 l+1 − l
Xk=1{vi,Xvj}∈E′
yi=wk
yj=wl
l<k
h
= (w2 w2 )
k − k−1
Xk=1{vi,Xvj}∈E′vXj∈/Vk
vi∈Vk
h
δ(V ) ∂V (w2 w2 )
≥ k | k| k − k−1
k=1
X
h
δ(H)i(H) V (w2 w2 )
≥ | k| k − k−1
k=1
X
= δ(H)i(H)( V (w2 w2 )+...+ V (w2 w2))
| h| h − h−1 | 1| 1 − 0
= δ(H)i(H)(( V V )w2 +...+( V V )w2)
| h|−| h−1| h | 1|−| 2| 1
n
= δ(H)i(H) y2
i
i=1
X
(y +y )2 = 2 (y2+y2) (y y )2
i j i j − i − j
{vi,Xvj}∈E′ vi,Xvj∈E′ vi,Xvj∈E′
n
2 d(v )y2 (y y )2
≤ i i − i − j
Xi=1 vi,Xvj∈E′
n
2∆(H) y2 (y y )2
≤ i − i − j
Xi=1 vi,Xvj∈E′
b
= 2∆(H) M 2∆(H) M.
− ≤ −
b
δ(H)2i(H)2 i(H)2
M ,
≥ 2∆ M ≥ 2∆ M
− −
solving which we get M ∆ √∆2 i2. Substituting in (7) we get
≥ − −
s
α min(∆ √∆2 i2).
≥ m − −
(cid:3)
10 ANALYTIC CONNECTIVITY IN GENERAL HYPERGRAPHS
4. Summary
In this paper we have built upon the results for analytic connec-
tivity of k-graphs given in [6] by applying the definitions for general
hypergraphs found in [1]. We have obtained bounds for the analytic
connectivity of general hypergraphs with respect to diameter and de-
gree sequence of the hypergraph. We also proved a version of Cheeger’s
inequality for hypergraphs.
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Department of Computer Science and Automation, Indian Institute
of Science, Bangalore 560012, India.
E-mail address: [email protected]